Porisms

And, lastly, when it is said, if a circle ABC be given in position, and also a point E, a point D may be found, such that if two lines EF and I'D be inflected from E and D to any point whatever F, in the circumference, these lines shall have a given ratio to one another : the proposition becomes a porism, and is the same we have just investi gated.

The algebraical method for the investigation of porisms, may very readily be deduced from the consideration of the definition which Mayfair has given, and the facilities which such a method presents in the discovery of this class of truths, is another instance of the advantages which result from a condensed method of expressing the relations of quantity. It has been stated that a jiorism is a proposition affirming the possibility of finding the indeterminate case of a problem.

If, therefore, any problem is proposed in which the quantity sought is called .r ; by means of the given condi tions some equation will be found between x and known quantities, which may be reduced to the form A+B.r+C +...-1-Nxn=0 (a) A, B,.. . being known functions of the constant quantities; from this equation x may be determined, or at least it cannot generally have more than a certain determinate number of values; such values of x satisfy the conditions of the problem ; but in order to discover whether any poris matic case exists, we must examine whether the data of the problem admit of such a relation amongst themselves, that we may have at the sante time A If this is the case, the equation (a) is verified indcpendenth of any particular value of x; and, instead of a limited, we have an indefinite number of solutions. This principle may be stated more generally thus, if a, b, c, .. are given quantities, and, x, y, z, . . . those which are to be found ; then the solution of the problem leads to several equations of the form (.r, y, . a, b, . .)=

If any relative which can be established amongst the constants, a, b, c, shall cause the equation which results from the elimination of the unknown quantities from these equations to be independent of any of them, then by sup posing that relation to exist we have a porism.

As an example, let us take the following problem. Sup pose a circle (Plate CCCCLXVII. Fig. 4.) whose radius is r, and a point C in its diameter, such that OC = v, also a straight line, FL, perpendicular to this diameter; let it be required to find the angle which the chord, PQ, must make with the diameter ; so that if another chord, P,Q,, be drawn at right angles to it, and from the extremities of these chords perpendiculars be drawn to the given line, the rectangle under those let fall from one chord on this line shall be equal to that under the perpendiculars let fall on the same line from the other chord. Let the required angle PCF = 0, then CE = v cos. 0 and OE = v sin. 0 Hence CP = v — sin. — v cos. 0 CQ = v — sin. + v cos. 9. Also CG cos. 0 a—v Therefore PG = + v cos. 0 — v sin. Bz cos.

a—v QG = cos. A + v cos. / r — sin. These multiplied by cos. A produce respectively PL = a — v + v cos. cos. cos. A v — cos. QM = a —v + v cos. COS. 0 — cos. Az The rectangle under these two lines is (a — v v cos. — cos. (ra sin. And this by proper reductions becomes PL. QM = (a — v + a v — v. cos. — In order to find the rectangle of the perpendiculars let fall from the extremities of another chord at right angles to this, we have only to change 0 into 2 — + A in this expres sion, and since cos. + 0) = sin. A we must substitute 1 — cos. instead of cos. which gives